Solve g_{1}(x)=1/x+1/x^2+1/x^5

$$g _ { 1 } ( x ) = \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } + \frac { 1 } { x ^ { 5 } }$$

Evaluate

$$\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{5}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x^{2} is x^{2}. Multiply \frac{1}{x} times \frac{x}{x}.

$$\frac{x}{x^{2}}+\frac{1}{x^{2}}+\frac{1}{x^{5}}$$

Since \frac{x}{x^{2}} and \frac{1}{x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x+1}{x^{2}}+\frac{1}{x^{5}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and x^{5} is x^{5}. Multiply \frac{x+1}{x^{2}} times \frac{x^{3}}{x^{3}}.

$$\frac{\left(x+1\right)x^{3}}{x^{5}}+\frac{1}{x^{5}}$$

Since \frac{\left(x+1\right)x^{3}}{x^{5}} and \frac{1}{x^{5}} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+1\right)x^{3}+1}{x^{5}}$$

Do the multiplications in \left(x+1\right)x^{3}+1.

$$\frac{x^{4}+x^{3}+1}{x^{5}}$$

Factor

$$\frac{x^{4}+x^{3}+1}{xx^{4}}$$